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There are 100 units of good 1 and good 2 in a economy. Consumer 1 and consumer 2 have 50 units of each good. Consumer 1 only wants good 1 whereas consumer 2 only wants good 2.

Note: Neither of these are lexicographic preferences.

Question: Find Walrasian equilibria

Here is what I have reached:

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Can someone explain why red lines are WE..and why not F and G? And P is Pareto optimal I understand, but is it Walrasian equilibrium as well?

Is every Walrasian equilibrium Pareto optimal and every Pareto optimal point Walrasian equilibrium?

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We have two Consumers 1 and 2, and two goods 1 and 2 pure exchange economy. The following utility functions can be used to represent their preferences:

  • $u_1(x_{11}, x_{12}) = x_{11}$
  • $u_2(x_{21}, x_{22}) = x_{22}$

Equilibrium price vector $(p_1, p_2=1)$ and allocation $((x_{11}, x_{12}), (x_{21}, x_{22}))$ satisfy the following:

Optimality Conditions (Allocation must solve the utility maximization problem of the two consumers, i.e. it must lie on their demand functions)

  • $(x_{11}, x_{12}) = \left(\frac{50p_1 + 50}{p_1}, 0\right) $
  • $(x_{21}, x_{22}) = \left(0, 50p_1 + 50\right)$

Market Clearing Conditions

  • $x_{11} + x_{21} = 100$
  • $x_{12} + x_{22} = 100$

Solving the above gives price vector $(p_1, p_2) = (1, 1)$ that supports the allocation $((x_{11}, x_{12}), (x_{21}, x_{22})) = ((100, 0), (0, 100))$ in equilibrium. This allocation is the only competitive equilibrium. It is also the only Pareto efficient allocation in this economy.

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